Annals of Physics 350 (2014) 50–68

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Enhancing robustness of multiparty quantumcorrelations using weak measurementUttam Singh a,∗, Utkarsh Mishra a, Himadri Shekhar Dhar ba Quantum Information and Computation Group, Harish-Chandra Research Institute, Chhatnag Road,Jhunsi, Allahabad 211 019, Indiab School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India

h i g h l i g h t s

• Extension of weak measurement reversal scheme to protect multiparty quantum correlations.• Protection of multiparty quantum correlation under local amplitude damping noise.• Enhanced fidelity of quantum teleportation in one sender and many receivers setting.• Enhanced fidelity of quantum information splitting protocol.

a r t i c l e i n f o

Article history:Received 17 April 2014Accepted 3 July 2014Available online 14 July 2014

Keywords:Weak measurement reversalMultiparty quantum correlationQuantum information splitting

a b s t r a c t

Multipartite quantum correlations are important resources for thedevelopment of quantum information and computation protocols.However, the resourcefulness of multipartite quantum correla-tions in practical settings is limited by its fragility under deco-herence due to environmental interactions. Though there existprotocols to protect bipartite entanglement under decoherence,the implementation of such protocols for multipartite quantumcorrelations has not been sufficiently explored. Here, we studythe effect of local amplitude damping channel on the generalizedGreenberger–Horne–Zeilinger state, and use a protocol of optimalreversal quantum weak measurement to protect the multipartitequantum correlations. We observe that the weak measurementreversal protocol enhances the robustness of multipartite quan-tum correlations. Further it increases the critical damping valuethat corresponds to entanglement sudden death. To emphasizethe efficacy of the technique in protection of multipartite quan-tum correlation, we investigate two proximately related quantumcommunication tasks, namely, quantum teleportation in a one

∗ Corresponding author.E-mail addresses: uttamsingh@hri.res.in (U. Singh), utkarsh@hri.res.in (U. Mishra), dhar.himadri@gmail.com (H.S. Dhar).

http://dx.doi.org/10.1016/j.aop.2014.07.0130003-4916/© 2014 Elsevier Inc. All rights reserved.

U. Singh et al. / Annals of Physics 350 (2014) 50–68 51

sender, many receivers setting and multiparty quantum informa-tion splitting, through a local amplitude damping channel. Weobserve an increase in the average fidelity of both the quantumcommunication tasks under the weak measurement reversal pro-tocol. Themethodmay prove beneficial, for combating external in-teractions, in other quantum information tasks using multipartiteresources.

© 2014 Elsevier Inc. All rights reserved.

1. Introduction

Quantum correlation is an intrinsic aspect of quantum theory that enables the manifestation ofseveral interesting phenomena beyond the realms of the classical world. The quintessential form ofquantum correlation is entanglement which is an important resource in various quantum informa-tion and computational protocols [1]. The fundamental need for scalability of quantum informationprocessing and computation protocols requires the generation of controlled quantum correlationsdistributed over large number of subsystems [2]. In particular, multipartite quantum correlation is anindispensable resource in one-way quantum computing [3], secret-sharing protocols [4,5] and quan-tum communication [6]. Recent experimental developments in quantum mechanics have enabledthe generation of small multipartite entangled states to simulate one-way quantum computers [7],graph states [8] and open-destination quantum teleportation [9], using trapped ions [10] and photons[7–9,11].

The practical realization of quantum information and computation protocols using multipartyquantum systems is severely challenged due to decoherence caused by the interaction of the sys-tem with the environment. Such interactions create superfluous quantum correlations between thesystem and the environment leading to information being scattered in the intractable Hilbert space ofthe environment. Therefore, the fragility of multipartite quantum correlations makes the generationand preservation of quantum correlations in any quantum system a daunting task for experimental-ists. From a theoretical point of view, the lack of a unique characterization of multipartite quantumcorrelations in quantum systems obstructs definitive study of decoherence-induced loss of correla-tions [12]. The decoherence models used to study the robustness of multipartite states, are mostly lo-cal; i.e., each subsystemof the state interacts independentlywith the environment [12–15]. Therefore,in order to successfully implement a quantum protocol, the degradation of the quantum correlations,through local decoherence channels, must be suppressed during the application of the protocol.

Various schemes such as distillation protocols [16] and quantum error correction [17] havetraditionally been used to protect entanglement under local decoherence (cf. [18]). The efficiencyof these methods depends on the robustness of the entanglement in the initial state. A more recentapproach to tackling local decoherence is using the quantum zeno effect [19] and the reversibilityof quantum weak measurement [20]. Other techniques for entanglement and quantum correlationpreservation have been addressed using structured environments [21] and classical driving randomfields [22]. The experimental viability of implementing quantum weak measurements makes thelatter method an elegant approach to counter decoherence. The suppression of decoherence on asingle qubit using the reversal of weakmeasurement has been experimentally exhibited [23]. Further,the protection of bipartite entanglement in two-qubit systems using a reversal weak measurementscheme has been studied [24] and experimentally demonstrated [25]. The method has also beenextended to include two-qutrit quantum correlations [26] and few-body quantum communicationtasks [27]. A natural progression of the weakmeasurement approach is to consider the suppression ofthe degradation of multipartite quantum correlations under local decoherence as they are desirablefor applications in scalable quantum information and computation protocols. An obvious difficultyin designing a weak reversal multipartite decoherence-protection scheme is the simultaneous

52 U. Singh et al. / Annals of Physics 350 (2014) 50–68

characterization of the quantum correlation measure and the weak measurement technique in themultipartite setting.

We use the local amplitude damping channel (LADC) as our decoherence model, and study its ef-fect on the generalized Greenberger–Horne–Zeilinger (gGHZ) state [28]. Such a channel produces amixed state of the system from an initial multipartite pure entangled state, rendering proceduralcharacterization of pure state multipartite quantum correlations irrelevant. Hence, we characterizethe mixed state multipartite quantum correlations using a multipartite extension of logarithmic neg-ativity [29] and a measure of global entanglement called the Mayer–Wallach measure [30]. We thenapply a multipartite generalization of the decoherence-protection protocol based on the reversibilityof weakmeasurement.We observe that the protocolmakes themultipartite quantum correlations ro-bust against LADC. This is also evident from the enhanced critical damping value that corresponds toentanglement sudden death. To further elucidate the efficiency of the protocol, we investigate quan-tum teleportation [31] in a one sender, many receivers setting and multiparty quantum informationsplitting [5,32] using the amplitude-damped gGHZ state and observe an increase in the average fidelityof teleportation and information splitting under the weak measurement reversal protocol.

The paper is organized in the followingway. In Section 2, we discuss theweakmeasurement rever-sal protocol to be applied to the initial multipartite state under consideration. The characterization ofthe multipartite quantum correlation measure is done in Section 3. The results for the suppression ofdecoherence and enhancement of multipartite quantum correlation is shown in Section 4. Section 5discusses the results of the quantum teleportation and quantum information splitting tasks. We con-clude in Section 6 with a brief discussion of the results and its ramifications.

2. Weak measurement reversal protocol

Weak measurements were initially developed for pre-selected and post-selected ensembles ofquantum systems byAharonov et al. [33] and later generalized to caseswithout post-selection [34]. Analternative definition of a weak measurement is obtained from partial collapse measurement [20]. Insuch instances, weak measurements are positive-operator valued measure with limited informationaccess as compared to projective measurement thus allowing the operation to be non-unitarilyreversed. The concept of reversing a partial collapse measurement was initially introduced in [35]and subsequently applied to quantum error-correcting codes [36]. In the field of superconductingqubits, weak measurement have been used to propose interaction-free measurements [37]. In recenttimes, a weak measurement reversal scheme has been developed to suppress the decoherence in asingle qubit state [20,23], and subsequently expanded to protect quantum correlations in two qubitstates [24–26]. The underlying principle for the weak measurement reversal scheme is the fact thatany partial collapse measurement can be reversed [36,38].

The weak measurement reversal for protection of multipartite quantum correlations can be de-scribed by the following protocol. Firstly, a weak measurement of a given strength, say s, is madeon the initial multiparty entangled state before the state is subjected to local decoherence. For ann-qubit initial entangled state, the measurement consists of n single-qubit weak measurements, eachof strength s, acting locally and independently on each qubit. Secondly, the post weak measured stateundergoes decoherence via a local amplitude damping channel, with damping parameter p, acting in-dependently on each qubit. Finally, the decohered state is subjected to a reversal weak measurement,with an optimal reversal strength, r = r0, that is attained by maximizing the multipartite quantumcorrelations. Again, the reversal weak measurement consists of n single-qubit reversal weak mea-surements, each of strength r = r0, acting locally and independently on each qubit. The protocol isgraphically illustrated in Fig. 1.

In the following parts, we discuss the amplitude damping channel, the weak measurement, andthe weak measurement reversal.

2.1. Amplitude damping channel

Amplitude damping channel is a very useful model of decoherence for studying various quantumphenomena such as spontaneous emission in quantum optics, energy dissipation in quantum

U. Singh et al. / Annals of Physics 350 (2014) 50–68 53

Fig. 1. Illustration of the weak measurement reversal protocol. The preparator prepares an initial n-qubit gGHZ state. It thenapplies local weak measurements of strength s on all individual qubits of the n-party state. The weak measured qubits arethen individually passed through a LADC with amplitude damping parameter p. After local amplitude damping, each qubit issubjected to a local reversal weak measurement of strength r . The multipartite quantum correlation of the final state is morerobust than an n-qubit state passing through LADC without weak measurement reversal protocol.

open-systems and capacities of quantum channels. The decoherence due to amplitude dampingresults in the coupling of a qubit, locally, to its environment leading to irreversible transfer of a basis tothe other. The effect of a LADC on a densitymatrix of a single qubit in computational basis is describedby the following map:

ε(p) : ρ → ε(p)(ρ) =

ρ00 + pρ11

1 − pρ01

1 − pρ10 (1 − p)ρ11

, (1)

where p is the amplitude damping parameter and ρij (i, j = 0, 1) are the elements of ρ in thecomputational basis. Thus, the amplitude damping channel keeps the computational basis state |0⟩⟨0|unchanged but transfers the state from |1⟩⟨1| to |0⟩⟨0| with probability p. The effect of a LADC ofstrength p on an n-qubit state, say ρn, is given by the map ε(p)⊗n(ρn).

2.2. Weak measurement

The weak measurement, used in our protocol is a positive operator valued measure, consisting ofa set of positive operators {Mi(s)} (i = 0, 1), such that

M0(s) =√s|1⟩⟨1|,

M1(s) = |0⟩⟨0| +√1 − s|1⟩⟨1|. (2)

If we discard the outcome of the measurement M0(s), then the measurement M1(s) is a null-resultweak measurement of strength s, that partially collapses the system to one of the basis states. The

54 U. Singh et al. / Annals of Physics 350 (2014) 50–68

action of the weak measurement operatorM1(s) on a qubit density matrix in computational basis canbe mapped as

Λ(s) : ρ → Λ(s)(ρ) =

ρ00

√1 − sρ01√

1 − sρ10 (1 − s)ρ11

, (3)

where ρij (i, j = 0, 1) are the elements of ρ in the computational basis. The quantum map of weakmeasurement is different from the LADC as it uses post-selection to selectivelymap the states. The dis-carded detections, via some ideal detector, ensures that the operator M1(s) keeps the computationalbasis state |1⟩⟨1| unchangedwith a probability (1− s) and a norm less than unity. Such post-selectionin weak measurements can increase or decrease quantum correlations in bipartite systems [39]. Theaction of the local quantum weak measurement, of strength s, on an n-qubit state, say ρn, is given bythe mapΛ(s)⊗n(ρn).

2.3. Reversal weak measurement

The weak measurement can be reversed using nonunitary operations. For every null-result weakmeasurement given in Section 2.2,we can obtain a nonunitary reverse operationN0(r)with strength r ,thatwill produce the initial statewith someprobability. Such anoperation is termed as a reversalweakmeasurement. Let us consider a measurement using the set of positive operators {Ni(r)}(i = 0, 1),such that

N0(r) = |1⟩⟨1| +√1 − r|0⟩⟨0|,

N1(r) =√r|1⟩⟨1|. (4)

Again if we selectively discard the outcome of measurement N1(r), then N0(r) constitutes a reversalweak measurement with strength r . For some optimized reversal weak measurement strength r =

r0, corresponding to the original weak measurement strength s, one can generate the pre-weakmeasurement state with some probability. The action of the weak measurement operator N0(r) ona qubit density matrix in computational basis can be mapped as

Φ(r) : ρ → Φ(r)(ρ) =

(1 − r)ρ00

√1 − rρ01√

1 − rρ10 ρ11

, (5)

where ρij (i, j = 0, 1) are the elements of ρ in the computational basis. Similar to the weak measure-ment, in the reversal measurement, the discarded detections ensures that the operator N0(r) keepsthe computational basis state |0⟩⟨0| unchanged with a probability (1− r) and a norm less than unity.The action of the local reversal quantumweak measurement of strength r on an n-qubit state, say ρn,is given by the mapΦ(r)⊗n(ρn).

3. Multipartite quantum correlations

In our study, the amplitude damping channel, the weak and the reversal measurements act locallyon each subsystem (qubit) of the gGHZ state. The LADC, in effect, renders the initial pure gGHZ stateto amixed quantum state. The study of multipartite quantum correlations in this decohered quantumstate will require the characterization of a multipartite measure. Though there are known measuresof multipartite entanglement [40] which can be computed in large pure quantum states [41], thesemeasures do not uniquely generalize to the mixed state regime. In the following parts we attempt todiscuss multipartite quantum correlations that shall prove useful for our study.

3.1. Multiparty logarithmic negativity

Logarithmic negativity (LN) is a useful and computable measure of entanglement introduced byVidal andWerner [29] based on the Peres–Horodecki criteria [42]. The criteria notes that the negativity

U. Singh et al. / Annals of Physics 350 (2014) 50–68 55

of the partial transpose of any bipartite state is a sufficient condition for entanglement along thatparticular cut. For an arbitrary bipartite state, ρA:B, LN is defined as

ELN(ρA:B) = log2ρTA

A:B

1

≡ log2[2N (ρA:B)+ 1], (6)

where N (ρA:B) = (1/2)(∥ρTAA:B∥1 − 1) is called the ‘‘negativity’’, and ∥ρ

TA12∥1 is the trace norm of ρTA

12 ,which is the partially transposed state of the bipartite state ρA:B with respect to the subsystem A. Thenegativity, N (ρA:B), is thus the sum of the absolute values of the negative eigenvalues of ρTA

A:B.For the gGHZ state, the negative partial transpose criteria of entanglement along all possible

bipartite cuts is a sufficient condition. Hence, for gGHZ states, LN is a measure of multipartiteentanglement [43,44].

3.2. Global entanglement

A scalable and functional entanglement monotone for multiqubit pure states, called the globalentanglement, was introduced by Meyer andWallach [30]. The Meyer andWallach measure of globalentanglement (MW) is closely related to the per qubit total nonlocal information [45,46] in the systemandhence can be easily related to the loss of nonlocal information per site under decoherence [47]. Thedistribution of nonlocal information in a system is known to be related to quantum correlations in amultipartite quantumsystem [48].MWmeasure is, thus ameasure of pure statemultipartite quantumcorrelations. Interestingly, the connection between the total nonlocal information and tangles can beused to characterize global entanglement in the decohered gGHZ state. The MW measure of globalentanglement, for an n-qubit state in terms of tangles, can be written as [47]

EMWgl =

1n

2

i1<i2

τi1i2 + 3

i1<i2<i3

τi1i2i3 + · · · + n

i1<···<in

τi1...in

, (7)

where τi1i2 is the 2-tangle, τi1i2i3 is the 3-tangle and τi1...in is the n-tangle. The term on the right side ofEq. (7), with a multiplicative factor (n), is the expression for total nonlocal information of a quantumsystem in terms of the tangles.

The k-tangle for an n-qubit state can be obtained by generalizing the idea of 3-tangle [49]. For thestates belonging to gGHZ class, the n-tangle (k = n) is the only nonzero tangle and, therefore, thesole contributor to Eq. (7) [50]. For multiparty states with even n, the n-tangle can be expressed interms of the square of a quantity called the n-concurrence [51], which is a suitable generalization ofconcurrence [52] for a two qubit state. Hence, for certain mixed multipartite states the MW measureof global entanglement can be exactly computed in terms of its n-concurrence, where n is even. Amathematical disposition of the MW measure for the weak measurement reversal protocol is givenin Section 4.

4. Multipartite quantum correlations under weak measurement reversal protocol

In this section, we apply the weak measurement reversal protocol to a local amplitude dampedn-qubit gGHZ state and study its effect on the decay of multipartite quantum correlations. Toinvestigate the effectiveness of the protection protocol, we study the decay of these measures fordifferent values of amplitude damping parameter p and system size n. We consider the n-qubitgeneralized GHZ (gGHZ) state [28] as our initial multipartite entangled state, which can be writtenas

|ψ⟩GHZ = α|0⟩⊗n+ β|1⟩⊗n, (8)

where α = cos(θ/2) and β = sin(θ/2)with 0 ≤ θ ≤ π . To apply the local decoherence in the initialstate, we pass each of the n qubits of the gGHZ state, separately, through the amplitude damping

56 U. Singh et al. / Annals of Physics 350 (2014) 50–68

channel, given in Section 2.1. The action of the amplitude damping channel on the initial gGHZ stateyields the following decohered n-qubit state

ρGHZ = ε(p)⊗n(|ψ⟩⟨ψ |GHZ) = |α|2(|0⟩⟨0|)⊗n

+ pn/2[αβ∗(|0⟩⟨1|)⊗n+ h.c.]

+ |β|2

nk=0

pkp(n−k)[(|0⟩⟨0|)⊗k

⊗ (|1⟩⟨1|)⊗n−k+ R], (9)

where p = 1−p.R denotes all other diagonal terms that can be obtained by permutations of positionsof 0 and 1 in the computational basis.

4.1. Using multiparty logarithmic negativity

We first consider, the calculation of themultipartite logarithmic negativity (Section 3.1). To obtainthe multipartite LN, we need to obtain the partial transpose of the state ρGHZ across all possiblebipartite cuts. Let us consider the m|(n − m) cut for partial transposition. The state after partialtransposition along this cut is given by

ρPTGHZ = |α|

2(|0⟩⟨0|)⊗n+ pn/2

αβ∗| 0 . . . 0

m

1 . . . 1 n−m

⟩⟨1 . . . 1 m

0 . . . 0 n−m

| + h.c.

+ |β|

2n

k=0

pkp(n−k)[(|0⟩⟨0|)⊗k

⊗ (|1⟩⟨1|)⊗n−k+ R], (10)

where the only transformation occurs in the non-diagonal terms of ρGHZ. The partial transposedmatrixρPTGHZ contains a diagonal block of dimension (2n

− 2) × (2n− 2) that is positive semi-definite. The

negative eigenvalue(s) can be obtained from the remaining 2 × 2 block, which corresponds to basisvectors {|0 . . . 0⟩m⊗|1 . . . 1⟩n−m, |1 . . . 1⟩m⊗|0 . . . 0⟩n−m}. The 2×2 non-positive semi-definite blockcan be written as

BρPTGHZ =

pmp(n−m)

|β|2 αβ∗pn/2

α∗βpn/2 pn−mpm|β|2

. (11)

The eigenvalue of the above block which contributes to the entanglement, is given by

ϵm =|β|

2

2

b(p, n,m)−

b2(p, n,m)+ 4pn

|α|2

|β|2− pn

, (12)

where b(p, n,m) = pmp(n−m)+ pn−mpm. Using the negative eigenvalue obtained from Eq. (12), we

calculate the multipartite LN (defined by Eq. (6)). The condition for nonzero entanglement, i.e., ϵm tobe negative is given by p < min{1, ( |α|

|β|)2/n}. Thus, the entanglement of the decohered gGHZ state

in the bipartition m|(n − m) vanishes for p ≥ pc , where pc = min{1, (|α|/|β|)2/n} is the criticaldamping value. pc is found to be independent of the bipartition used to calculate the LN. For |α| = |β|

case, pc = 1, which is the case of maximum decoherence. However, numerically the entanglementapproaches values close to zero, before the critical value pc = 1. In our analysis, we define a newcritical value of p, denoted by pac , as the least value of p above which the entanglement is less thanor equal to 10−3. Interestingly, for the case of |α| < |β|, the multipartite LN becomes zero before themaximal value of decoherence parameter, p = 1, is attained. This phenomenon is called entanglementsudden death (ESD) [53].

Now let us consider the operations under the weak measurement reversal protocol, as mentionedin Section 2. As discussed earlier, we first apply a weak measurement (Λ(s)⊗n), of strength s, on theinitial n-qubit gGHZ state. We then pass the weak measured state through an amplitude dampingchannel (ε(p)⊗n), with damping parameter p. Finally, we perform a reversal weak measurement(Φ(r)⊗n), of strength r , to obtain the final state.

U. Singh et al. / Annals of Physics 350 (2014) 50–68 57

The normalized gGHZ state after the application of a weak measurement of strength s, is given by

ρwGHZ =Λ(s)⊗n

[ρGHZ]

Tr[Λ(s)⊗n[ρGHZ]]

= |α1|2(|0⟩⟨0|)⊗n

+ |β1|2(|1⟩⟨1|)⊗n

+ [α1β∗

1 (|0⟩⟨1|)⊗n

+ h.c.], (13)

where α1 = α/√

T1, β1 = sn/2β/√

T1 and s = 1 − s. T1 = |α|2+ sn|β|

2 is the success probability ofweakmeasurement. Theweakmeasured state is thenpassed through a LADCwith damping parameterp that acts locally on each qubit. The decohered state is given by

ρwGHZ = |α1|2(|0⟩⟨0|)⊗n

+ pn/2[α1β∗

1 (|0⟩⟨1|)⊗n

+ h.c.]

+ |β1|2

nk=0

pkp(n−k)[(|0⟩⟨0|)⊗k

⊗ (|1⟩⟨1|)⊗n−k+ R1], (14)

where again, R1 denotes all other diagonal terms that can be obtained either by permutations ofpositions of 0 and 1 in the computational basis. Then we make a reversal weak measurement ofstrength r , on the state ρwGHZ. The state after the reversal weak measurement is given by

ρwrGHZ =

1T2

|α1|

2 rn(|0⟩⟨0|)⊗n+ (r p)n/2[α1β

∗

1 (|0⟩⟨1|)⊗n

+ h.c.]

+ |β1|2

nk=0

(pr)kp(n−k)[(|0⟩⟨0|)⊗k

⊗ (|1⟩⟨1|)⊗n−k+ R1]

, (15)

where

T2 = |α1|2 rn + |β1|

2n

k=0

nCk(pr)kpn−k= |α1|

2 rn + |β1|2(1 − pr)n, (16)

and r = 1 − r . T2 is the success probability of the reversal weak measurement. The transmissivity orthe success probability of the final weak measurement reversal protocol, is given by

T = T1T2 = |α|2 rn + |β|

2sn(1 − pr)n. (17)

Let us again consider the m|(n − m) bipartition for the partial transposition of the final postreversal weakmeasurement state ρwr

GHZ. Following the steps for partial transpose of the non-protectedamplitude damped state we again obtain a 2× 2 matrix block that contains the possible non-positiveeigenvalue. The matrix form of such a block can be found to be

Bρwr(PT )GHZ

=1T2

(pr)mpn−m

|β1|2 α1β

∗

1 (pr)n/2

α∗

1β1(pr)n/2 (pr)n−mpm|β1|2

. (18)

The eigenvalue, which can be negative is

ϵwrm =

|β1|2

2T2

b1(r, p,m, n)−

b21(r, p,m, n)+ 4(r p)n

|α1|

2

|β1|2

− pn, (19)

where b1(r, p,m, n) = (pr)mp(n−m)+ (pr)(n−m)pm. The condition for nonzero entanglement, i.e.,

ϵwrm to be negative is given by p < min{1, ( 1s )(

|α|

|β|)2/n}. This shows that the weak measurement

reversal protocol enhances the critical damping value of p as compared to the unprotected case. Thus,a higher damping parameter p is required to remove entanglement. The optimal value of the reversalweak measurement strength r is obtained by maximizing the multipartite LN for a fixed value of thedamping strength p, the number of qubits n, the initial weak measurement strength s and all possiblebipartitions m. We will denote this optimal value of LN by Eopt

LN . For |α| < |β| and specific values of

58 U. Singh et al. / Annals of Physics 350 (2014) 50–68

(a) n = 4. (b) n = 8.

(c) n = 12. (d) n = 24.

Fig. 2. (Color online) The optimal multipartite logarithmic negativity of the amplitude-damped gGHZ state with α = β =

1/√2, under the protection protocol, as function of amplitude damping parameter p, for different values of initial weak

measurement strength s, system size n and at the bipartition m = n/2. We observe that the decay of EoptLN is retarded and

the critical value of p where correlation death occurs is increased for finite values of s. The plot label UP is for the unprotectedstate (s = r = 0).

Fig. 3. (Color online) The critical damping value, p = pac , for EoptLN plotted against the number of qubits n. The plot shows that

pac of the protected state remains above the unprotected state for n = 100, though the difference diminishes with increasing n.The plot label UP is for the unprotected state (s = r = 0).

the weak measurement strength s, the observed ESD under the amplitude damping can be suitablycircumvented.

Fig. 2 represents the optimal value of the multipartite LN (EoptLN ) of the amplitude-damped gGHZ

state (with α = β = 1/√2), under the weak measurement reversal protocol, as a function of

the damping strength p, for various initial weak measurement strength s, and for n qubits, at thebipartitionm = n/2. We observe that the weak measurement reversal protocol suppresses the decayof multipartite quantum correlations, for different n. From Fig. 2, we observe that Eopt

LN decreases withincrease of the amplitude damping parameter p and vanishes after a critical damping value pac , thatdepends on n. For smaller values of n (n = 4 and n = 8), the weak measurement reversal techniqueleads to a substantial increase in themultipartite entanglement for larger values of s, making the statemore robust against thenoise. However,weobserve later that the arbitrary high values of s ∈ [0, 1] arenot allowed. At higher values of n, say n = 24, it is apparent from Fig. 2 (also see Fig. 3) that pac becomes

U. Singh et al. / Annals of Physics 350 (2014) 50–68 59

(a) p = 0.2. (b) p = 0.3.

Fig. 4. (Color online) Transmissivity as a function of the initial weakmeasurement strength s for different values of n and fixeddamping p. We observe that the success probability reduces for larger number of qubits n.

smaller and the weak measurement reversal protocol does not sufficiently enhance entanglement(due to limitation on s), rendering the state less robust against decoherence for very large values of n.

Fig. 3 shows the variation of pac with n, for fixed values of the weak measurement strength s. Weobserve that the values of pac for all s, get closer at higher values of n, which implies that enhancementof the critical value for protected states is lower, rendering the protocol less efficient at very large n.

The effective value of the weakmeasurement strength s also depends on the success probability ofthe reversal protocol. To obtain the success probability of the protocol we calculate its transmissivity,defined in relation (17). For an initial gGHZ state with α = β = 1/

√2, the transmissivity is given by

the expression T =12 [r

n+ sn(1 − pr)n]. The transmissivity T is calculated for the optimal value of

the reversal weak measurement strength r for which ELN is maximal, i.e. ELN = EoptLN .

We observe from Fig. 4, that the success probability reduces with increase in the weak measure-ment strength s and hence, the protocol becomes less robust. Thus, for larger values of the initialweak measurement strength, the success probability of the quantum state reversal using weak mea-surement becomes lower and hence the number of initial copies of the system needed to successfullyimplement the protection protocol increases.

4.2. Using global entanglement

We again consider the n-qubit gGHZ state given by Eq. (8). For the MW measure of global entan-glement (EMW) in Eq. (7), it is known that the multipartite quantum correlation for the n-qubit gGHZstate is contained only in the n-tangle part of the nonlocal information, since all the other k-tangles(k = n) are zero [50]. For an even number of qubits (even n), the n-tangle can be calculated using then-concurrence measure Cn, as shown in [51]. Cn is a generalization of two qubit measure of concur-rence [52]. Therefore for an initial, even n-qubit gGHZ state the MW global entanglement is given bythe relation

EMW =

i1<···<in

τi1...in = τn = C2n ,

where Cn is defined as

Cn = max

0,

λ1 −

2Ni=2

λi

, (20)

where λi’s are the eigenvalues of the matrix Rn = ρσy⊗nρ∗σy

⊗n (for any density matrix ρ) in decreas-ing order, and σy is the Pauli matrix. The gGHZ state, after passing through local amplitude dampingchannel is given by Eq. (9). The eigenvalues of Rn = ρGHZ(ρ

scGHZ) are given by

λ1 = |β|2pn

(2|α|

2+ |β|

2pn)+

4|α|2(|α|2 + |β|2pn)

,

60 U. Singh et al. / Annals of Physics 350 (2014) 50–68

λ2 = |β|2pn

(2|α|

2+ |β|

2pn)−

4|α|2(|α|2 + |β|2pn)

,

λj = |β|4(pp)n, {j = 3, 4, . . . , 2n

}, (21)

where ρscGHZ is the spin conjugated state obtained after applying σy⊗n on ρ∗

GHZ. Therefore, the n-concurrence is given by

Cn(ρGHZ) = max

0,

λ1 −

λ2 −

2nj=3

λj

= max

0,

2|α| |β|pn/2

1 − (2n−1

− 1)|β|

|α|pn/2

. (22)

The above expression shows that the n-concurrence is zero for p ≥ pMWc , where pMW

c =

min{1, ( |α|

|β|(2n−1−1))2/n}, and positive definite otherwise. Hence ESD occurs at the critical damping

parameter value, p = pMWc . In the limit of very large n, the critical value pMW

c becomes 1/4,independent of the parameters of the initial gGHZ state. The n-concurrence indicates that the gGHZcan sustain decoherence up to p = 0.25 for very large n, unlike the logarithmic negativity which,in principle can sustain decoherence up to p = 1. The global entanglement, EMW of the state ρGHZ isgiven by C2

n (ρGHZ). We define the numerical critical value of p in the case of EMW, pMWac , as the value

of p below which the EMW is less than 10−3.Now the state after the weak measurement reversal protocol is given by (15). The eigenvalues of

R′n = ρwr

GHZ(ρwr(sc)GHZ ) are given by

λ′

1 =|β1|

2(pr)n

T 22

(2|α1|

2+ |β1|

2pn)+

4|α1|

2(|α1|2 + |β1|

2pn),

λ′

2 =|β1|

2(pr)n

T 22

(2|α1|

2+ |β1|

2pn)−

4|α1|

2(|α1|2 + |β1|

2pn),

λ′

j =|β1|

4(ppr)n

T 22

, {j = 3, . . . , 2n}, (23)

where ρwr(sc)GHZ is the spin conjugated state obtained after applying σy⊗n on ρwr∗

GHZ . The n-concurrence is,then, given by

Cn(ρwrGHZ) = max

0,

λ′

1 −

λ′

2 −

2nj=3

λ′

j

= max

0,

2|α1| |β1|(r p)n/2

T2

1 − (2n−1

− 1)×|β1|

|α1|pn/2

= max

0,

2|α| |β|(sr p)n/2

|α|2 rn + |β|2sn(1 − pr)n

1 − (2n−1

− 1)|β|

|α|(sp)n/2

. (24)

The expression for Cn(ρwrGHZ) shows that the n-concurrence is zero for p ≥ pMW

c , where pMWc =

min{1, 1s (

|α|

|β|(2n−1−1))2/n}, and positive definite otherwise. In the limit of very large n, the critical value

of pMWc becomes 1/(4s), which is a function of initial weak measurement strength s but independent

of the parameters of the initial gGHZ state. The n-concurrence of the protected gGHZ can sustaindecoherence up to p = 0.25/s for very large n. This indicates that the protected state is morerobust against decoherence and ESD occurs at higher amplitude damping values, as compared to theunprotected state.

U. Singh et al. / Annals of Physics 350 (2014) 50–68 61

(a) n = 4. (b) n = 8.

(c) n = 12. (d) n = 24.

Fig. 5. (Color online) The global entanglement of the gGHZ state with α = β = 1/√2 as function of amplitude damping

parameter p for different values of initial weak measurement strength s and system size n. We observe that the decay of EoptMW

is retarded and the critical value of p, pMWac is increased for increasing values of s. The plot label UP is for the unprotected state

(s = r = 0).

The global entanglement, EMW is given by C2n (ρ

wrGHZ). The optimal value of the global entanglement,

EoptMW, after the weakmeasurement reversal protocol can be obtained by numerically maximizing EMW,

with respect to the weak reversal strength r , at fixed values of n, s, and p. Fig. 5 shows the optimalvalue of global entanglement, Eopt

MW, against the amplitude damping parameter p, for different values ofn and s. The plots show that Eopt

MW increases with weak measurement strength s. Again, arbitrary largevalues of s in [0, 1] are limited, as it will lead to low success probability of the weak measurementreversal protocol. It is also clear from the plots that the critical value pMWa

c is always greater for theweak measurement applied state compared to the unprotected state. We observe that the value ofpMWac is not suitably enhanced for very large n (see Fig. 6).

5. Quantum teleportation in one sender, many receivers setting and multiparty quantuminformation splitting

In Section 4, we have seen that multipartite entanglement can be protected against LADC, usingweak measurement reversal protocol. The immediate question that arises, is whether such a protec-tion scheme can prove beneficial in some quantum communication task? Herewe consider, two prox-imately related quantum communication tasks, namely, quantum teleportation of an unknown qubitstate in a one sender, many receivers setting [31,5] and multiparty quantum information splitting[5,32], that, under decoherence, are known to be associatedwith different aspects ofmultiparty quan-tum correlations [54].We consider the decohered gGHZ state as the shared resource in these quantumcommunication tasks and calculate the average fidelities under the weak measurement protectionprotocol.

According to the one sender, many receivers quantum teleportation protocol, Alice (A) wants tosend an unknown qubit in a state (|ψ0⟩ = α|0⟩ + β|1⟩) to (n − 1) Bobs (Bi, i = 1–n − 1) using ashared n-qubit GHZ state, given as

|φ⟩A:B1..Bn−1 =1

√2[|0⟩⊗n

+ |1⟩⊗n]. (25)

The success of a quantum teleportation protocol can be evaluated by the average fidelity between theunknown initial qubit to be teleported and the final qubit received during the process. The fidelity of

62 U. Singh et al. / Annals of Physics 350 (2014) 50–68

Fig. 6. (Color online) The critical damping value, p = pMWac , for Eopt

MW is plotted against the number of qubits n. The plot showsthat the critical value, pMWa

c , of the protected state remains above the unprotected state for n = 100, though the differencediminishes with increasing n. The plot label UP is for the unprotected state (s = r = 0).

teleportation is defined as F = ⟨ψ0|ρ|ψ0⟩, where |ψ0⟩ is the unknown initial state and ρ is the finalteleported state. For a perfect quantum teleportation protocol, with unit fidelity, A and Bi’s must sharea maximally quantum correlated state [31,55]. The maximum fidelity achievable using only classicalcommunication, with no quantum correlations, is Fcl = 2/3 [56].

For the considered n-qubit GHZ state, given by (25), the pre-shared amount of quantum correlationis sufficient to teleport the unknown state from Alice to (n − 1) Bobs with maximum fidelity. Toelaborate, Alice makes a Bell measurement on the unknown qubit, |ψ0⟩, and the qubit (A) of then-qubit pre-shared GHZ state, in her possession, and classically communicates her results to (n − 1)Bobs. The information to be teleported is encoded in the reduced (n−1)-qubit state in Bobs possession.On receiving Alice’s result, the (n − 1) Bobs together apply suitable local unitaries to obtain thestate

|ψf ⟩ = α|0⟩⊗(n−1)+ β|1⟩⊗(n−1). (26)

Applying local measurements, the (n − 1) Bobs together can distill the unknown state, |ψ0⟩ =

α|0⟩ + β|1⟩, with unit fidelity, from the encoded state, |ψf ⟩.However, for practical applications, the pre-shared resource is generally not optimal for perfect

teleportation. Let us consider a slightly non-idealistic situation where some arbitrary quantum statepreparator prepares the n-qubit gGHZ state, given by Eq. (25), and sends the qubits to Alice and Bobs.Since Alice and Bobs are at distant locations, the state sent by the preparator suffers from decoherencedue to environmental interaction. Such a decohered state can be modeled by sending the preparedn-qubit gGHZ state through n single-qubit LADCs (Section 2.1) to Alice and Bobs. Thus, the state sharedby Alice and (n−1) Bobs is given in Eq. (9) (withα = β = 1/

√2). The average fidelity of teleportation

using such a shared state is known to decrease with increasing damping strength p [54].In the weak measurement reversal protocol, the preparator first applies a weak measurement of

strength s on each shared qubit of the state. These weak measured states are then sent to Alice and(n−1) Bobs through the local amplitude damping channel, with strength p. Now, Alice and (n−1) Bobslocally apply reversal weak measurements of equal strength r , to all the n qubits. The final measuredstate, using Eq. (15), is now the pre-shared quantum state, given by

ρwrA:B1..Bn−1

=12T

rn(|0⟩⟨0|)⊗n

+ (r ps)n/2[(|0⟩⟨1|)⊗n+ h.c.]

+ snn

k=0

(pr)kp(n−k)[(|0⟩⟨0|)⊗k

⊗ (|1⟩⟨1|)⊗n−k+ R1]

, (27)

Alice now wants to teleport the unknown qubit |ψ0⟩ to (n − 1) Bobs using the n-qubit pre-sharedstate ρwr

A:B1..Bn−1. The joint state is given by, |ψ0⟩⟨ψ0|⊗ρ

wrA:B1..Bn−1

. Alice now possesses a qubit (A) of the

U. Singh et al. / Annals of Physics 350 (2014) 50–68 63

n-qubit pre-shared state ρwrA:B1..Bn−1

and the unknown qubit |ψ0⟩ and makes a Bell basis measurementon the two qubits in her possession. The unknown state to be teleported is now encoded in somereduced (n − 1)-qubit state in (n − 1) Bobs possession. Let |φ±

⟩ =1

√2(|00⟩ ± |11⟩) and |ψ±

⟩ =

1√2(|01⟩±|10⟩) be the Bell basis states. Onmeasurement, Alice gets |φ±

⟩ as outcomeswith probability

p±

1 and the normalized state of (n − 1) Bobs is ρ±

1 , where

ρ±

1 =1

4p±

1 T

|α|

2 rn|0⟩⟨0|⊗(n−1)+ sn(|α|

2pr + |β|2p)

pr|0⟩⟨0| + p|1⟩⟨1|

⊗(n−1)

± (r ps)n/2αβ∗

|0⟩⟨1|⊗(n−1)+ h.c.

B1..Bn−1

, (28)

and

p+

1 = p−

1 =|α|

2 rn + sn(|α|2pr + |β|

2p)(pr + p)n−1

4T. (29)

Similarly, Alice gets |ψ±⟩ as outcome with probability p±

2 and the normalized state of (n − 1) Bobs isρ±

2 , where

ρ±

2 =1

4p±

2 T

|β|

2 rn|0⟩⟨0|⊗(n−1)+ sn(|β|

2pr + |α|2p)

pr|0⟩⟨0| + p|1⟩⟨1|

⊗(n−1)

± (r ps)n/2αβ∗

|1⟩⟨0|⊗(n−1)+ h.c.

B1..Bn−1

, (30)

and

p+

2 = p−

2 =|β|

2 rn + sn(|β|2pr + |α|

2p)(pr + p)n−1

4T. (31)

Now depending on themeasurement outcomes of Alice, the (n−1) Bobs apply suitable local unitarieson the encoded state, ρi± (i = 1, 2), to obtain the optimal (n − 1)-qubit state, say, ρj (j = 1–4),with corresponding probabilities pj = {p±

1 , p±

2 }, that maximizes the fidelity of the teleportation. Fromrelation (26), it is known that in the absence of LADC, the encoded state |ψf ⟩, allows (n − 1) Bobsto distill the unknown state, |ψ0⟩, with unit fidelity. Hence, the success of the protocol, depends onthe fidelity of (n − 1) Bobs optimal encoded state, ρj, with the state |ψf ⟩. Thus, the average fidelity ofteleportation can be calculated using the relation, ⟨F ⟩tel =

j pj⟨ψf |ρj|ψf ⟩.

Using the weak measurement reversal protocol, the average fidelity (⟨F ⟩tel) of quantum telepor-tation is given by the expression

⟨F ⟩tel =16T

2rn(1 + pnsn)+ 2pnsn + sn(prpn−1

+ p(pr)n−1)+ 2(r ps)n/2. (32)

The average fidelity of teleportation is maximized with respect to the reversal weak measurementstrength r , keeping the parameters p, s, and n fixed. Without the protection protocol (r = 0 ands = 0), the average fidelity is given by the expression (2 + pn−1(1 + p)+ pn−1(1 + p)+ 2pn/2)/6, asobtained in [54]. Fig. 7 is the plot for the numerically optimized average fidelity of quantum telepor-tation. It can be observed from the figure that the critical value pfc(n) (beyond which the optimizedaverage fidelity is less than or equal to the classical teleportation fidelity, 2/3) has increased with theinitial weakmeasurement strength s, as compared to the average fidelity without any weakmeasure-ment protection. It is also observed from Fig. 7 that the optimal average fidelity always remains aboveor equal to classical fidelity that is ⟨F ⟩tel ≥ Fcl, for finite weak measurement reversal application.

In the one sender, many receivers quantum teleportation protocol, the average fidelity ofteleportation is obtained by maximizing the fidelity of the encoded (n− 1)-qubit states in possessionof Bobs with the state |ψf ⟩. The (n − 1) Bobs perform as a single entity to optimize the outcome.An alternate approach is to consider the multiparty quantum information splitting protocol [5,32] todecode the unknown state, |ψ0⟩. In this protocol, the (n − 1) Bobs are independent, but co-operative,

64 U. Singh et al. / Annals of Physics 350 (2014) 50–68

(a) n = 4. (b) n = 8.

(c) n = 12. (d) n = 24.

Fig. 7. (Color online) The average fidelity of teleportation (⟨F ⟩tel) as a function of amplitude damping parameter p fordifferent values of initial weak measurement strength s and number of qubits n. We observe that the noisy channel underweakmeasurement reversal protocol never reduces below the classical resource limit by always generating an average fidelity,⟨F ⟩tel ≥ 2/3. We also observe that the critical damping value leading to the classical limit, under amplitude-damping, isincreased for finite s. At p = 1, the average fidelity of teleportation for unprotected state becomes 2/3, which we have notshown for the sake of brevity. The plot label UP is for the unprotected state (s = r = 0).

entities that can classically communicate with each other. A specific Bob (say (n−1)th Bob) is chosenas the pre-determined receiver with the other (n − 2) Bobs assisting it to receive the unknown state,|ψ0⟩. The information is thus being split amongmany receivers and can be retrieved only by necessaryco-operation. This forms the basis of many secret sharing protocols [5].

In the quantum information splitting protocol, the encoding of the state to be teleported by Aliceremains the same as the quantum teleportation protocol. Alicemakes a Bell basismeasurement on thetwo qubits in her possession, the qubit (A) of the n-qubit pre-shared state ρwr

A:B1..Bn−1and the unknown

qubit |ψ0⟩. After Alice’s measurement, the encoded states, ρj = {ρ1±, ρ2

±} (j = 1–4), given by

Eqs. (28) and (30), with corresponding probabilities pj = {p±

1 , p±

2 }, are split among the independent(n − 1) Bobs. The (n − 2) non-receiver Bobs measure their respective qubits in the eigenbasis of thePauli spin matrix σx and classically communicate their outcomes, which occur with probabilities, qjk(j = 1–4, k = 1–2n−2), to the (n − 1)th Bob. Depending on the measurement outcomes of Alice andthe (n − 2) Bobs, (n − 1)th Bob applies suitable unitary operations to obtain the single-qubit state,say, σ j

k (j = 1–4, k = 1–2n−2), with probability pjqjk. The average fidelity of the protocol can then

be calculated using the relation, ⟨F ⟩is =

j,k pjqjk⟨ψ0|σ

jk|ψ0⟩. For a maximally entangled pre-shared

state, such as the n-qubit GHZ state, the (n − 1)th Bob can obtain the unknown state, |ψ0⟩, with unitfidelity.

Under the action of LADC and weak measurement reversal protocol, the average fidelity ofteleportation using multiparty quantum information splitting (⟨F ⟩is), is given by the expression

⟨F ⟩is =13T

rn + sn(p + pr)n−2(p2 + p2 r2 + ppr)+ (r ps)n/2

. (33)

The average fidelity using information splitting can be maximized with respect to the reversal weakmeasurement strength r , keeping the parameters p, s, and n fixed. Without the protection protocol(r = 0 and s = 0), ⟨F ⟩is is given by (2 − pp + pn/2)/3, as obtained in [54]. Fig. 8 is the plot for thenumerically optimized average fidelity of teleportation using quantum information splitting. It canbe observed from the figure that the critical value pfc(n) (beyond which the optimized average fidelityis less than or equal to the classical teleportation fidelity, 2/3) has increased with the initial weak

U. Singh et al. / Annals of Physics 350 (2014) 50–68 65

(a) n = 4. (b) n = 8.

(c) n = 12. (d) n = 24.

Fig. 8. (Color online) The average fidelity of teleportationusing information splitting (⟨F ⟩is) as a function of amplitudedampingparameter p for different values of initial weak measurement strength s and number of qubits n. We observe that the noisychannel under weak measurement reversal protocol never reduces below the classical resource limit by always generating anaverage fidelity, ⟨F ⟩is ≥ 2/3. We also observe that the critical damping value leading to the classical limit, under amplitude-damping, is increased for finite s. The plot label UP is for the unprotected state (s = r = 0).

measurement strength s, as compared to the average fidelity without any protection from the weakmeasurement reversal protocol. It is also observed from Fig. 8 that the optimal average fidelity alwaysremains above or equal to classical fidelity that is ⟨F ⟩is ≥ Fcl, for finite weak measurement reversalapplication.

We observe that both the communication tasks, viz., one sender, many receivers quantumteleportation and multiparty quantum information splitting, have greater average fidelity under theweak measurement reversal protocol, when the pre-shared resources suffer from decoherence. Thefundamental difference between the two communication tasks is in the optimization to obtain thedesired outcome. While the first protocol requires joint optimization of the encoded states by (n− 1)Bobs to allow maximal distillation of the outcome, the second protocol uses multiparty informationsplitting to decode the state, with the pre-determined Bob performing the optimization. It is arguedin [54], that the two protocols utilize different aspects of multipartite quantum correlations to get theaverage fidelity above the classical upper bound, using W-like state as a resource. However, in ourinvestigation, we observe that the two quantum communication tasks, using a decohered multiqubitgGHZ state, do not exhibit any relation with a specific kind of multiparty entanglement measure. Forexample, in the absence of the weak measurement protection protocol, starting with a four qubitdecohered gGHZ state, we observe that the MW measure goes to zero for p ≥ 1/

√7 but fidelity of

information splitting is greater than 2/3 for p ≥ 1/√7. This clearly shows that the MW measure

cannot be the only unique resource for the information splitting protocol (cf. [54]).

6. Conclusion

Quantum correlations are the basic resources for various quantum information protocols. Fromthe perspective of futuristic designs of quantum devices, the scalability of quantum resources is animportant aspect of contemporary research. As such, quantum correlations in multipartite quantumsystems need to be harnessed and generated. Hence, the study of decoherence protected quantumsystems are of fundamental and practical importance from the perspective of multipartite quantumcorrelations.

In our study, we have considered a multipartite, n-qubit gGHZ state undergoing local amplitudedamping and quantified its mixed state multipartite quantum correlations using logarithmic

66 U. Singh et al. / Annals of Physics 350 (2014) 50–68

negativity and global entanglement. We have then formulated a decoherence suppression andquantum correlation protection scheme based on weak measurement and reversal technique. Usinganalytical characterization and numerical optimization we have evaluated the multipartite quantumcorrelations under the weak measurement reversal protocol. We observe that under such a protocol,the multipartite correlations are more robust and do not vanish for low damping. In particular weobserve an enhanced value of the damping parameter corresponding to the ESD. To investigate theefficacy of the weak measurement reversal protocol, we study two quantum communication tasks,namely, quantum teleportation in a one sender, many receivers setting and quantum informationsplitting through local amplitude damping channel. We observe that the protocol enhances both theaverage fidelity of teleportation and information splitting and prevents the channel from performingbelow the classical upper-limit.

The weak measurement reversal protocol thus strengthens the robustness of global quantumcorrelations in multipartite systems under the effect of local amplitude damping. Given the factthat such weak measurements can be experimentally performed, the protection protocol may provepractically useful in enhancing performance in quantum information tasks, in presence of noise.

Acknowledgments

H.S.D. acknowledges University Grants Commission for financial support under the SeniorResearch Fellowship scheme. H.S.D. thanks theHarish-Chandra Research Institute (HRI) for hospitalityand support during visits. The authors thank Arun Kumar Pati for various helpful discussions. U.S.thanks Rafael Chaves and Yong-Su Kim for helpful communications.

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